Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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1answer
22 views

Can we deduce if a set is measurable, given a measurable function and a measurable space?

Let $f(x):X\rightarrow Y $, where $X$ is a measurable space. Suppose that $f$ is measurable. Let $E$ be a subset of $X$. Now, suppose that $f(E)$ is closed or clopen. Can we deduce that $E$ is a ...
0
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1answer
12 views

Why does a Hermitian operator with singleton spectrum have to be scalar?

One proof of Schur's lemma proceeds by showing that a Hermitian intertwining operator of an irreducible representation (of a topological group on a Hilbert space) has a spectrum that contains only one ...
2
votes
2answers
62 views

A property for an ODE

$2\leq n\in\mathbb{N}$. I have no idea how to show that there is a unique solution $y\in C^1([0,T))$ of the ODE \begin{eqnarray} \begin{cases} y'(t)=(1+y(t)^2)\left(1-\dfrac{n-1}{t}y(t)\right)\ \ \ ...
0
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1answer
23 views

Are maximal intervals of open nonempty sets always equal?

Let $O\subset\mathbb{R}$ be an open nonempty interval. Define for every $x\in O$: $$a_x = \inf\{a\in\mathbb{R}\mid(a,x]\subset O\}$$ $$b_x = \sup\{b\in\mathbb{R}\mid[x,b)\subset O\}$$ $$I_x = (a_x, ...
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0answers
9 views

Consistent estimators/convergene in probability and slutsky

Let $m_n$ be a consistent estimator of $g(\vec\alpha)$ where $\vec\alpha = (\alpha_1,\cdots,\alpha_k)\in \mathbb{R}^k$ and $v_n$ be a consistent estimator of $f(\alpha_1,g(\vec\alpha))$. Suppose that ...
1
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1answer
45 views

Let $A,B \subset \mathbb{R}^n$ open sets, $A\subset B$. How can I set a function $f:\mathbb{R}^n\longrightarrow\mathbb{R}$, $C^{\infty}$

Let $A,B \subset \mathbb{R}^n$ open sets, $A\subset B$. How can I set a function $f:\mathbb{R}^n\longrightarrow\mathbb{R}$, $C^{\infty}$, such that: \begin{equation*} f(x)=0, \textrm{ if } x\in ...
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3answers
49 views

Find an example of a sequence not in $l^1$ satisfying certain boundedness conditions.

This question is about getting a concrete example for this question on bounded holomorphic functions posed by @user122916 (something that he really expected as explained in the comments). Give an ...
3
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1answer
21 views

Equivalency of the set of real numbers to the set of all continuous real functions?

I understand that the set of real numbers is equivalent to the set of real numbers in the interval $(0,1)$ and also equivalent to the set of all points in $\mathbb{R}^2$. I have seen a claim in a book ...
0
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1answer
21 views

Application of Residue theorem

Let f(z,w) be holomorphic in $\mathbb{C}^{n}$ and not identically zero on the w-axis. Let {$b_{j}$} be the set of singularities of f(z,w) in some disk of radius $|w| < r$. Why does the residue ...
3
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2answers
30 views

Equivalency of real numbers and points in the plane?

I understand that the set of real numbers is equivalent to the set of real numbers in the interval $(-1,1)$ by simply using $arctan$ function. However, I do not know how to find a one-to-one mapping ...
1
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1answer
15 views

define $f(\mathbf{x})=f_1(x_1)+\cdots +f_n(x_n)$. Show that $f$ has a differential at each point of an n-dimensional interval.

Given $n$ real-valued functions $f_1, \dots, f_n$, defined and having finite derivatives in the interval $(a,b)$. For each $\mathbf{x}$ in the $n$-dimensional interval $$S=\{(x_1,\dots ,x_n)\mid a\lt ...
4
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0answers
37 views

$\phi_{\epsilon} \ast \mu \rightarrow \mu$?

Let $\phi$ be a non-negative function on $\mathbb{R}$ with $\int_{\mathbb{R}} \phi = 1$. Define $\phi_{\epsilon}(x)=\epsilon^{-1}\phi(\epsilon^{-1}x)$ for $x \in \mathbb{R}, \epsilon > 0$. For $f ...
2
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2answers
52 views

Homeomorphism between the set of invertible matrices and itself

Consider the set of invertible $n \times n$-matrices $GL_n(\mathbb{R}) = \{A \in M_{n \times n}(R) \mid A\text{ is invertible}\}$. I now want to prove that the transformation $$f: A \mapsto A^{-1}$$ ...
-1
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1answer
60 views

How can I find out if a non-convergent series is “indeterminate” (that is, “oscillating”) or “divergent”?

Definitions: Given a sequence $\{a_n\}$, define $$s_n= \sum_{j=0}^n a_j.$$ The sequence $\{s_n\}$ is called the series of partial sums of $\{a_n\}$. A series is convergent if $\{s_n\}$ has ...
3
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1answer
35 views

Conformal maps onto open right half plane

On the Big Rudin there is the conformal map $$\varphi(z) = \frac {1+z}{1-z}$$ which sends $\{-1, 0, 1\}$ to $\{0, 1, \infty\}$. The book says: The segment $(-1, 1)$ maps onto the positive real ...
0
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1answer
24 views

Laplacian operator on $L^2(\Omega)$?

Let $\Omega\subseteq \mathbb R^n$ be an open subset and $\displaystyle \Delta:=-\sum_{j=1}^n D_j^2$ be the Laplacian operator. I have some questions concearning this operator: $(i)$ Does it map ...
5
votes
1answer
87 views

$f, \hat{f} \in L^{p}(\mathbb R) \cap C(\mathbb R) \implies |f(x)| \to 0$ as $|x| \to \infty$?

Suppose $f, \hat{f} \in L^{p}(\mathbb R) \cap C(\mathbb R)\cap L^{\infty}(\mathbb R), (1<p<\infty).$ My Question: Can we expect $\lim_{|x|\to \infty} |f(x)|=0$ ? (In other words, If $f$ and its ...
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2answers
33 views

Study: $\sum_{n=1}^\infty (\sin(\sin n))^n$, $\sum_{n=1}^\infty \frac{n \sin (x^n)}{n + x^{2n}}$, and $\sum_{n=1}^\infty \frac{ \sin (x^n)}{(1+x)^n} $

Let $x \in \mathbb{R}$. I have to study the convergence of the following three series: $$\sum_{n=1}^\infty (\sin(\sin n))^n$$ $$\sum_{n=1}^\infty \frac{n \sin (x^n)}{n + x^{2n}}$$ ...
1
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0answers
23 views

Cauchy Criteria for Series

We know that the Cauchy Criterion of a series is as follow: Theorem: A series $\sum\limits_{i=1}^{\infty}x_i$ converges iff for all $\epsilon>0$ there is an $N\in \mathbb{N}$ so that for all ...
0
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1answer
24 views

operator norm of a linear transformation, given by the transformation matrix

Consider $\mathbb{K}^n$, $\mathbb{K}^m$, both with the $||.||_1$-norm, where $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$. Let $||T|| = inf\{M ≥ 0: ||T(x)|| ≤ M ||x|| \space \forall x \in ...
0
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1answer
16 views

Additive function in $\mathbb{R}^n$ is continuous, and related subspaces compact

I want to show that the function: $A: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n, (x, y) \mapsto x + y$ is continuous. Also, why is it that if $K, L$ are compact subspaces of $\mathbb{R}^n$, ...
5
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4answers
83 views

Constructing the natural numbers without set theory.

As I understand it the idea of defining everything as sets is a relatively new idea in mathematics. Does that mean there's a non-set theoretic definition of the natural numbers? Could there be?
2
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0answers
43 views

For which values of $\alpha, \beta, x \in \mathbb{R}, x \geq 0$, does the series $\sum_{n=1}^\infty n^ \alpha x^{n^{\beta}}$ converge?

I have to study, for $\alpha, \beta, x \in \mathbb{R}$, $x \geq 0$, the convergence/divergence/irregularity (i.e., when the limit of the $N$-th partial sum does not exist) of the following series: ...
0
votes
0answers
30 views

Factorization of the sine

I am working on the Basel problem for a project for my Mathematics study. I need to proof that one could write the sine as a factorization of its linear roots. I know the proofs is in general done bye ...
1
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1answer
20 views

Locality of tensors part of definition?

I am wondering whether linearity with respect to scalar functions $f \in C^{\infty}(M, \mathbb{R})$ is part of the definition of a tensor? Let me explain it by referring to the Riemann curvature ...
3
votes
2answers
64 views

how can I show this integral diverges?

I want to show $E(T_a)=\infty$ $$E(T_a)=\int_0^{\infty}{{x|a|}\over\sqrt{2\pi}}x^{-3/2}e^{-a^2/x}dx$$ to show this I need to show this integral diverges. I know gamma function that $$\Gamma ...
1
vote
0answers
25 views

Proof of Weierstrass Preparation Theorem

In Griffiths and Harris, Principles of Algebraic Geometry, on page 8, near the end of the proof of the Weierstrass Preparation Theorem, he states that $h(z,w)$ has only removable singularities in the ...
1
vote
1answer
33 views

Difference of support functions and its minimum points

Let $A$ and $B$ are convex, compact sets in $\mathbb{R}^n$. We have known that $$\max_{a\in A}\min_{b \in B} \|a-b\|=\sup_{\|g\|\le1}(\sigma_A(g)-\sigma_B(g)),$$ where $\sigma_M(x)=\sup_{u\in ...
2
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1answer
38 views

Easy application of the Riemann Mapping Theorem

Riemann Mapping theorem Every simply connected region $\Omega \subset \mathbb C$ is conformally equivalent to the open unit disk (except $\Omega = \mathbb C$) What are application of this ...
0
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0answers
22 views

Determine the order of $\exp Q(z)$ when $Q$ is a polynomial of degree $q$.

I am looking to determine the order of $f(z) = \exp Q(z)$ when $Q$ is a polynomial of degree $q$. I think the order is $q$, but I am struggling to prove it. The definition of order is: An entire ...
0
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1answer
28 views

Proof of the Harnack inequality

Let $\Omega\subseteq\mathbb{R}^n$ be a domain, $\Omega'\subset\subset\Omega$ be a domain and $u\in C^0(\overline{\Omega})$. Suppose we know $$\sup_{\Omega'}u\le 3^n\inf_{\Omega'}u\tag{1}$$ if ...
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0answers
22 views

Convergent sequences doubt from Rudin

What does the following mean: The sequence $<1/n>$ converges in $\mathbb R$ (to 0) but fails to converge in the set of all positive real numbers [with $d(x,y) = |x-y|$]. Reading Rudin's ...
2
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2answers
30 views

Does $\lim\limits_{(x,y)\to (0,0)}\frac{x^4}{y}$ exist?

Does $\lim\limits_{(x,y)\to (0,0)}\frac{x^4}{y}$ exist? I don't think so, therefore I try to find a sequence $(x_n,y_n)\to(0,0)$ such that the limit of $f(x_n,y_n)$ in $(0,0)$ does not exist. Do you ...
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votes
1answer
65 views

Is Cantor set closed? [closed]

https://www.youtube.com/watch?v=dazO9UoKmyA This professor said Cantor set is closed because it's FINITE union of closed intervals at 14.00. But isn't it a wrong statement since Cantor set is ...
1
vote
1answer
33 views

Showing if $f$ unbounded, then $f$ is not Riemann Integrable

I want to show that if $f:[a,b] \to \mathbb{R}$ is unbounded, then $f$ is not Riemann Integrable. I suppose that $f$ is unbounded above: $$\forall_{n \in \mathbb{N}} \exists_{t_n} \in [a,b], \; ...
1
vote
4answers
56 views

Prove or disprove $\frac{\left(2^{p}-2\right)}{p}\ \in \Bbb N, \forall\, p,\, prime$

Apologies in advance for poor formatting, not completely accustomed to typeset. What I ask is any non-particular value p, with one condition that it is prime, for which to disprove the following ...
0
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1answer
19 views

convergence of $\int_{x=0}^{\infty}x^{-\frac{M-1}{2}-N}(1-e^{-x})^{M-1}e^{-x}dx$

I am trying to study the convergence of $$\int_{x=0}^{\infty}x^{-\frac{M-1}{2}-N}(1-e^{-x})^{M-1}e^{-x}dx,$$ where $M$ and $N$ are positive integers. I've tried some $M$ and $N$, and it seems that ...
0
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0answers
33 views

Uniform convergence implies $L^2$ convergence

Let $f_n : [a,b] \rightarrow \mathbb{R}$ where $n \in \mathbb{N}$ be a sequence of functions. We say $f_n \rightarrow f$ uniformly if $\forall \epsilon > 0$ $\exists N \in \mathbb{N}$ such that ...
0
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0answers
39 views

Does this set have content zero? Is it Jordan Measurable?

Does the set $\mathbb{Q} \times [0,1]$ have content zero in $\mathbb{R}^2$? Is it Jordan measurable? We say that a set $H$ has content zero if $\forall \epsilon > 0$ $\exists$ boxes ...
0
votes
1answer
28 views

How to find out if a sequence with exponentiation in fraction is convergent

I need to find the convergence of this function: $\sum^{\infty}_{x=1}{\frac{(x+1)^{x^2}}{x^{x^2}2^x}}$ Now my problem is, I have no clue how to do this (I tried the root-test and it did not work ...
6
votes
1answer
46 views

Is every element contained in a smallest measurable set?

Let $(X,\mathcal F)$ be a measure space, then for each $x \in X$ does there always exists a smallest measurable set containing $x$? If $X$ is countable or $\mathcal F$ finite, then this is true, as ...
4
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1answer
61 views

Poincaré Lemma, differential forms and I do have troubles

I think I need some hints about a proof I am currently reading in order to understand it. This question is similar to the construction used in Lemma 17.9 in the book "Introduction to smooth manifolds" ...
0
votes
1answer
22 views

Limsup of product of sequences

A sequence $\langle X_n\rangle\rightarrow x > 0$ as $n \rightarrow \infty$; $\limsup(Y_n) = y$ as $n \rightarrow\infty$. Show that $\limsup(X_nY_n) = xy$ as $n \rightarrow\infty$. My thinking: ...
2
votes
1answer
27 views

Inversion map is a Conformal map

I'm studying PDE by Evans book and I need to show that the inversion map $f:\mathbb{R}^n-\{0\}\to \mathbb{R}^n$, defined by $$f(x)=\frac{x}{\|x\|^2}$$ is conformal. So I have a hint, show that ...
0
votes
1answer
21 views

show that $d(a,b)\leq r-s-t \Rightarrow K(c,t) \subseteq K(a,r)$ in a metric space, assuming that $r,s,t>0$ and $c \in K(b,s)$.

Alright, so in a metric space, $M$, with $r,s,t>0$, $a,b,c \in M$ and $c \in K(b,s)$ I have to show, that: $d(a,b)\leq r-s-t \Rightarrow K(c,t) \subseteq K(a,r)$ I really have no idea where to ...
1
vote
1answer
13 views

ratio between volumes in $\mathbb{R}^n$

Let $[-a_n,a_n]^n$ be the largest cube that fits into the n-sphere $S^{n-1}.$ Can we say what $a_n$ is? I mean, for $n=1$ we have $a_1=1$ and for $n=2$ we have $a_2 = \frac{1}{\sqrt{2}},$ so does ...
1
vote
1answer
54 views

Evaluate $\int_{x=0}^{\infty}\left(\frac{1}{\sqrt{x}}(1-e^{-x})\right)^{M-1}e^{-x}(1+sx)^{-N}dx,$

I am trying to evaluate $$\int_{x=0}^{\infty}\left(\frac{1}{\sqrt{x}}(1-e^{-x})\right)^{M-1}e^{-x}(1+sx)^{-N}dx,$$ where $s>0$, $M$ and $N$ are positive integers. But seem that the above integral ...
3
votes
0answers
47 views

An application of the uniform boundedness principle

Can someone provide me with a sketch of proof / hint for this exercise: Let K $\subset$ $L^1([0,1]\,,\, \mu)$ be a closed linear subspace. If $\forall$ $f \in K$, $\exists \, p > 1$ such that $f ...
1
vote
2answers
59 views

Example 2, Sec. 11.1 in Apostol's CALCULUS, vol. 1: How to calculate this limit?

For each $n \in \mathbb{N}$, let $f_n \colon [0,1] \to \mathbb{R}$ be defined as $$f_n(x) \ \colon= \ nx (1-x^2)^n \ \mbox{ for all x } \ \in [0,1]. $$ Then $f_n(0) = 0 = f_n(1)$. So let $0 < x ...
2
votes
5answers
92 views

To evaluate $\lim_{n\to\infty} \dfrac{10^n}{n!} .$

I am having a lot of trouble evaluating $$\lim_{n\to\infty} \dfrac{10^n}{n!} .$$ I know intuitively that $n!$ is much larger and this expression should go to zero but I just can't figure out how to ...